User talk:12AbBa/TON analysis
Wrong Explanation > Remember that all ordinals can be expressed in TON. Therefore, there is a one-to-one correspondence: TON ordinals <--> Normal ordinals No way. p-adic 10:20, November 3, 2019 (UTC) ::Thought exactly the same thing, just a couple of minutes ago. While I was tryiing to figure out how to comment (forgot that these kind of pages have a talk page), you come and mention the same thing. Plain'N'Simple (talk) 10:27, November 3, 2019 (UTC) :: I first guessed that the OP comfounded "ordinal" in Taranovsky's sense (Taranovsky calls a TON expression an ordinal) with "ordinal" in set theory, but then the statement would be a tautology. Since the OP also state :: > Note that unlike in other notations, Ωn is not the nth uncountable. It is just a large ordinal representing fixed points. The nth CK ordinal will do, or the nth stable ordinal, or whatever. :: , I guess that the OP is making a mistake on the difference among the map assigning ordinal types and another map assigning a non-recursive ordinal to Ωn, which is actually confusing. :: p-adic 11:08, November 3, 2019 (UTC) :::: It seems like the statement you just quoted is an attempt to refer to a unique fact about TON: That unlike ordinary OCFs, the Ωn's in TON is just symbols. They are not actual ordinals at all. When we do analysis, we can map Ωn to an actual (probably nonrecursive) ordinal, but TON itself is just a system for manipulating and comparing symbols. :::: (granted, that's not what he actualy wrote. But I think that's the point he was going for) ::::The really confusing part, for many people, is that ordinary OCFs do afford similar mappings which leave them invariant. You can pretty much always replace Ωn (normally the nth cardinal) with the nth CK ordinal (or the nth inaccessible or many other variations), and it won't change the final result. ::::The only difference between TON and an actual OCF, is that an actual OCF needs some kind of "default values" for it do be well-defined. It needs actual ordinals to collapse. It doesn't really matter, what the specific values of the Ω's are (as long as they are far enough apart from one another). But we need to set them to something, because otherwise the entire structure becomes ill-defined. Plain'N'Simple (talk) :::: Of course, I understand the difference. Please remember that I am the one who wrote the blog post about the difference between OCFs and general ordinal notations, which are not necessarily derived from actual OCFs. :::: The point is that a TON expression such as Ωn is not literary an actual ordinal, and hence the statement that it can be any (sufficiently large) ordinal is literary false. (I know that we can assign larger cardinals to Buchholz's notation D_n 0 than usual Ω_n, but I guess that the OP is actually comfounding expressions with ordinals.) Such a confusion occurs when people identify an expression and its image of a suitable map. In the case of a general ordinal notation, we have a canonical map given by ordinal types, whose values are countable, and another (non-unique) map which assigns several symbols to be uncountable. Therefore I guessed that people somtimes misunderstand uncountable ordinals and countable ordinals through the identification of an expression with its values by distinct maps. (I could not imagine another reason why the OP regarded as the class of ordinals as a countable set.) :::: p-adic 12:42, November 3, 2019 (UTC) ::::::I know the you understand the difference. I was simply pointing out that this is probably what the OP meant, and also explained (not for you, but for others who might be reading this) what's the exact mathematical situation here is. :::::: ::::::By the way, for the vast majority of the people here, expressons are the same thing as ordinals. It isn't even a confusion of terms. It's just that nearly everybody on this wiki is using the word "ordinal" to refer to some intuitive concept rather than the formal set theoretic term. They couldn't care less about the set theoretic term, so it isn't even an error. :::::: ::::::For certain purposes, this isn't even a problem. You can climb quite high on the ordinal ladder without knowing any formal set theory. The problem begins at the OCF level, where we begin to heavily require set theoretic constructs to make sense. At this stage we are no longer just building bigger and bigger recursive structures (which is the part that can be done intuitively), but also borrowing concepts from the nonrecursive world to make our stuff work. And if a person borrows concepts he does not understand, it usually doesn't end well. Plain'N'Simple (talk) 15:01, November 3, 2019 (UTC) Hello, 12AbBa. Why are you keeping a wrong explanation even though it is already pointed out? Or couldn't you even understand the issue? p-adic 14:52, November 5, 2019 (UTC) Οkay! I will correct it.12AbBa (talk) 08:56, November 6, 2019 (UTC) : Xiexie, dansi you forgot the main issue: an ordinal does not necessarily correspond to a TON expression. It means that your explanation "Remember that all ordinals can be expressed in TON. Therefore, there is a one-to-one correspondence: TON ordinals <--> Normal ordinals" is wrong. : p-adic 09:38, November 6, 2019 (UTC) ::Oh you know Chinese. Also, can you give an example of an ordinal that does not correspond to TON? 12AbBa (talk) 13:51, November 6, 2019 (UTC) ::... Yeah. TON has a limit... 12AbBa (talk) 13:53, November 6, 2019 (UTC) ::: Dui, I am also a Chinese Googlogy Wiki user. And right, TON has a limit. Literary it can present only countably many ordinals, because there are only countably many sequences of C, 0, and Ω. Since there exist uncountably many ordinals, we have an ordinal which does not correspond to a TON expression. (This is related to the fact why Catching function defined by using TON is supposed to be ill-defined.) ::: p-adic 14:35, November 6, 2019 (UTC) ::: ::: Also, please address me by my real name, Zongshu. 12AbBa (talk) 09:53, November 7, 2019 (UTC) :::: Sure. :::: p-adic 12:24, November 7, 2019 (UTC) 0th system There is 0th system. It uses Ω0 and 0-built from below (a is 0-built from below from < b iff a < b). It has limit \(\Omega_0^{\Omega_0^{\Omega_0^\cdots}}=\varepsilon_1\). {hyp/^,cos} (talk) 14:23, December 15, 2019 (UTC) Really? Your webpage doesn't mention Ω0. Also, what is the value of Ω0? Is it ε0? Zongshu Wu 09:49, December 17, 2019 (UTC) Oh yeah now i understand. Zongshu Wu 10:04, December 17, 2019 (UTC)